Sunday, December 21, 2008
The basic idea is this. We know that the sine function has a well behaved Taylor series (it is an exponential function after all) so it makes sense to treat the sine function as the limit of a sequence of polynomials. The same logic works (as far as I can see) for any function with an absolutely convergent Taylor series.
Now polynomials have two basic representations. They can either be written as a sum of monomials like or they can be written as the product of factors like where the r's are the roots of the polynomial and A is just a (constant) scaling factor out front.
Now, assuming that the sine function has such a product representation the trick is to find A. We already know the r's: the roots of the sine function are simply the integer multiples of pi. So we expect thatand we simply need to find the constant A. To do this we simply need to evaluate both sides at a particular value of x. It could get difficult to do this though unless we pick a nice value for x. Let's try x=0. But both sides are zero when x is zero so this is no good! Solution: divide both sides by x and take the limit as x goes to zero. This is a common trick really, and it works a treat here.
Now we know A. All that remains is to put it together. But notice that there is one term in the denominator in A for each term in the product from sin(x)/x. We can take a typical term and simplify it like this . So the product ends up being:
So there you have the Wallis product formula for the sine function. From there it is easy to find a product formula for various multiples of pi etc by just plugging in values and rearranging to suite.
I realize that it is pretty straightforward maths and rather boring for most people probably. However I just found the beauty and simplicity of this product formula for such a common function worth sharing. I also realise that the LaTeX in this post is ugly and inconsistent but I have no good technique for rendering equations for this blog at the moment.
I hope that, with a touch of holiday spirit, you might forgive this silly maths nut. And maybe even find a little inspiration.
Tuesday, December 16, 2008
Conservation of Energy, Momentum, Angular Momentum, Electrical Charge, etc.
Conservation laws, in general, are very useful in that they tell us that certain quantities don't change. For instance, if you calculate the combined momentum of two objects before a collision you can be sure that the total momentum of the bodies after the collision will be the same provided there are no external forces acting on the system. This applies even if the nature of the bodies has changed considerably, for instance two cars deforming and breaking into pieces during a collision. Or say an electron and positron collide and annihilate, producing two photons. Momentum is still conserved even though the nature of the particles involved changes completely during the collision.
Conservation laws allow us to calculate the outcome of many processes (or at least severely restrict the range of possible outcomes) without having to understand the internal details of the process at all.
The ultimate origin of conservation laws is intimately related to the next principle...
Symmetry in Physical Law
Symmetry is the idea that there are operations that can be performed upon something, such that afterward the thing looks the same as it did before. Symmetry in physical law is then the idea that the mathematical equations of physics are invariant under certain groups of transformations. (Here group is a technical term for a set of algebraic objects which obey certain laws of combination. Group theory is the natural way to study symmetries mathematically.)
Symmetry in physical law appears at first glance an apparently empty concept. After all, the form of the laws doesn't change after the symmetry operation, so what does that tell us? Well, it turns out that, so far, every result of physics is derivable from a symmetry principle. This is truly incredible, and I think the implications of this have yet to filter through to the undergrad level, although the result itself has been known for many years.
Because of a mathematical result called Noether's theorem every (continuous) symmetry of physical law is directly related to a conservation law. For instance:
- The conservation of energy is a result of the symmetry of physical law under translations in time.
- The conservation of momentum is due to the symmetry of physical law under translations in space. This, incidentally, explains why energy and momentum are so closely related in relativity: it's just because space and time are so closely related in relativity.
- The conservation of angular momentum is due to the symmetry of physical law under rotations. The tricky bit here is that you can tell when you are rotating, but not that you have rotated. The rotations considered must be constant in time in order to get the right symmetry group.
- The symmetry of physical laws under "boosts" (changes of velocity of the reference frame) results in the conservation of a quantity which, as far as I'm aware, lacks a popular name but is nonetheless conserved. It equates to the law that the center of mass of a system moves uniformly in a straight line (in the absence of forces).
- The conservation of electric charge is related to an abstract (i.e., "purely mathematical") symmetry operation called U(1) gauge symmetry. It is not a symmetry like translations or rotations in space. It is a transformation of an abstract quantity called the phase of the wavefunction - a quantum mechanical thing that is hard to describe in words (although the mathematics is unambiguous). A truly remarkable thing happens when you try to quantize a theory involving a gauge symmetry, which will be discussed under the Quantum Field Theory heading.
- Other gauge transformations based on the SU(3)xSU(2)xU(1) symmetry group of the standard model result in conservation laws for quantities which are less well known, such as weak hypercharge and color charge etc.
A famous example of this is the once posited SU(2) iso-spin symmetry of hadrons. This resulted in a theory of the strong nuclear force that is only approximately accurate. However, as this theory is considerably simpler than the currently accepted theory (quantum chromodynamics) it is still sometimes used today as an approximation in various situations.
So the question is, how do we decide which symmetry groups to use in the description of nature? Answer: experiment.
Stay tuned for a follow up post including the action principle (my personal pick for most beautiful in all of physic), quantum field theory and possibly other things...
A burglar broke into a house one night. He shined his flashlight around, looking for valuables, and when he picked up a CD player to place in his sack, a strange, disembodied voice echoed from the dark saying, “Jesus is watching you.”
He nearly jumped out of his skin, clicked his flashlight off and froze. When he heard nothing more after a bit, he shook his head, promised himself a vacation after the next big score, then clicked the flashlight on and began searching for more valuables.
Just as he pulled the stereo out so he could disconnect the wires, clear as a bell he heard, “Jesus is watching you.”
Freaked out, he shined his light around frantically, looking for the source of the voice. Finally, in the corner of the room, his flashlight beam came to rest on a parrot. “Did you say that?” he hissed at the parrot…
“Yep”, the parrot confessed, then squawked, “I’m just trying to warn you.”
The burglar relaxed, then enquired, “Warn me, huh? Who the hell are you?”
“Moses,” replied the bird.
“Moses?”, the burglar laughed, “What kind of people would name a bird Moses?”
“The same kind of people that would name a Rottweiler, Jesus.”
Friday, December 12, 2008
I mean evolution may be wrong (let's look at the evidence), but even if it is wrong why do people assume then that God created everything as is a few thousand years ago?
Who says there only two options?
Wednesday, December 10, 2008
Tuesday, December 9, 2008
- Sunday school theology can only deal with Sunday school problems. (Yet very few people - believers and nonbelievers alike - attempt to go far beyond Sunday school theology.)
- God is bigger than any of my ideas about Him. (Yet most people think their ideas about God are way more important - and certain - than God himself.)
Finally, here is one of my favorite quotes:
Usually, even a non-Christian knows something about the earth, the heavens, and the other elements of this world, about the motion and orbit of the stars and even their size and relative positions, about the predictable eclipses of the sun and moon, the cycles of the years and seasons, about the kinds of animals, shrubs, stones, and so forth, and this knowledge he holds to as being certain from reason and experience. Now, it is a disgraceful and dangerous thing for an infidel to hear a Christian, presumably giving the meaning of Holy Scripture, talking nonsense on these topics; and we should take all means to prevent such an embarrassing situation, in which people show up vast ignorance in a Christian and laugh it to scorn. The shame is not so much that an ignorant individual is derided, but that people outside the household of the faith think our sacred writers held such opinions, and, to the great loss of those for whose salvation we toil, the writers of our Scripture are criticized and rejected as unlearned men.... Reckless and incompetent expounders of Holy Scripture bring untold trouble and sorrow on their wiser brethren when they are caught in one of their mischievous false opinions and are taken to task by these who are not bound by the authority of our sacred books. For then, to defend their utterly foolish and obviously untrue statements, they will try to call upon Holy Scripture for proof and even recite from memory many passages which they think support their position, although they understand neither what they say nor the things about which they make assertion.Source: St. Augustine (http://www.asa3.org/ASA/topics/Bible-Science/PSCF3-88Young.html)
Monday, December 8, 2008
Wife: Oh really?
Me: Yeah. You couldn't tell?
Me: Didn't think so.
I don't know what he did but he sure made a lot of noise. There are paint chips everywhere.
Cauchy Integral formula:
Lagrangian of quantum electrodynamics:
Not the most elegant option, but I suppose it works. The white against black is horrendous. Hrm... need to try something different.
- The computer turns itself off one day for no apparent good reason (not a blackout).
- It refuses to boot up again (Windows XP here). The file system is corrupted.
- System restore doesn't work.
- Neither does chkdsk or fsck or anything else I have on hand.
- Hard drive recovery is $$.
- Luckily I have a spare hard drive in a drawer somewhere. So I'll store the current drive safely away until I (we) decide what to do with it. I install the spare drive.
- It's only 4 GB. What does it have on it?
- Oh, an installation of Linux that also doesn't boot, inexplicably.
- Yaay, I still have some Linux distro discs lying around, I'll try installing them.
- Still won't boot.
- Well, a friend is coming up in a few days. I'll ask her to bring her XP cd with her.
- She does so. I install XP.
- XP boots, eventually. Oh, it's vanilla XP, not even SP1. Dang. My modem driver requires SP1 at least.
- I figure out a way to trick the modem driver install into thinking I have SP1, get online, then actually get SP1.
- Run out of disk space installing updates.
- Hmm... tweek the system a bit. Get a few hundred meg free space.
- Install more updates. Now I have SP2.
- Try installing useful programs, don't get many before running out of disk space.
- Tweek the system some more. Still not much space.
- Wait... I can use my USB memory stick as a (very slow) external hard drive!
- Now I have an office suite and web browser and some space for storage.
- The computer is usable for most things, although every few minutes I invariably get a "Critically Low Disk Space" warning. All of my old documents are gone, or, more precisely, on another hard drive with a fragged filesystem.
- Crossing fingers for a new hard drive (or new computer? *crosses harder*) this Christmas.
Sunday, December 7, 2008
I guess my goal here is to write down whatever comes into my head, whatever I think is neat or worth thinking about. Topics I consider interesting range from Christian theology, mathematical physics, astronomy, evolutionary biology, short fiction, current events, politics, video games...
I hope you get the point. Stay tuned.